Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $a = \dfrac{t^2 - 16t + 60}{t^2 - 7t} \div \dfrac{t - 10}{9t - 63} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{t^2 - 16t + 60}{t^2 - 7t} \times \dfrac{9t - 63}{t - 10} $ First factor the quadratic. $a = \dfrac{(t - 10)(t - 6)}{t^2 - 7t} \times \dfrac{9t - 63}{t - 10} $ Then factor out any other terms. $a = \dfrac{(t - 10)(t - 6)}{t(t - 7)} \times \dfrac{9(t - 7)}{t - 10} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ (t - 10)(t - 6) \times 9(t - 7) } { t(t - 7) \times (t - 10) } $ $a = \dfrac{ 9(t - 10)(t - 6)(t - 7)}{ t(t - 7)(t - 10)} $ Notice that $(t - 7)$ and $(t - 10)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ 9\cancel{(t - 10)}(t - 6)(t - 7)}{ t(t - 7)\cancel{(t - 10)}} $ We are dividing by $t - 10$ , so $t - 10 \neq 0$ Therefore, $t \neq 10$ $a = \dfrac{ 9\cancel{(t - 10)}(t - 6)\cancel{(t - 7)}}{ t\cancel{(t - 7)}\cancel{(t - 10)}} $ We are dividing by $t - 7$ , so $t - 7 \neq 0$ Therefore, $t \neq 7$ $a = \dfrac{9(t - 6)}{t} ; \space t \neq 10 ; \space t \neq 7 $